2009 AMC 10 B, Problem #21—“The sum of any four consecutive powers of 3 is divisibleby 8.”Difficulty:HardNCTMStandard:AlgebraStandard:understandandperformtransformationssuchasarithmetically combining, composing, and inverting commonly used functions, using technology toperform such operations on more-complicated symbolic expressions.Mathworld.com Classification:Number Theory>Divisors>Divisible
A cubical cake with edge length 2 inches is iced on the
sides and the top. It is cut vertically into three piecesas shown in this top view, whereMis the midpointof a top edge.The piece whose top is triangleBcontainsccubic inches of cake andssquare inches oficing. What isc+s?ABCM(A)245(B)325(C)8 +√5(D)5 +16√55(E)10 + 5√52009 AMC 10 B, Problem #22—“TriangleBis similar to triangleA.”SolutionAnswer (B):The area of triangleAis 1, and its hypotenuse has length√5. TriangleBis similar to triangleAand has a hypotenuse of 2, so itsarea is(2√5)2=45. The volume of the required piece isc=45·2 =85cubicinches.The icing on this piece has an area ofs=45+ 22=245squareinches. Thereforec+s=85+245=325.Difficulty:HardNCTM Standard:Geometry Standard: explore relationships (including congruence and similarity)among classes of two- and three-dimensional geometric objects, make and test conjectures aboutthem, and solve problems involving them.Mathworld.com Classification:Geometry>Plane Geometry>Triangles>Special Triangles>Other Triangles>Similar Triangles
Rachel and Robert run on a circular track.Rachelruns counterclockwise and completes a lap every 90seconds, and Robert runs clockwise and completes alap every 80 seconds. Both start from the start lineat the same time.At some random time between10 minutes and 11 minutes after they begin to run, aphotographer standing inside the track takes a picturethat shows one-fourth of the track, centered on thestarting line. What is the probability that both Racheland Robert are in the picture?(A)116(B)18(C)316(D)14(E)5162009 AMC 10 B, Problem #23—2009 AMC 12 B, Problem #18—“Find out the snapshot after 10 min of where Racheland Robert are.”16.Difficulty:HardNCTM Standard: